Noname manuscript No.(will be inserted by the editor)

Acceleration statistics of inertial particles in turbulent flow

Nauman M. Qureshi, Unai Arrieta, Christophe Baudet, Alain Cartellier, Yves Gagne, Mickael

Bourgoin

Laboratoire des Ecoulements Geophysiques et Industriels, CNRS/UJF/INPG UMR5519, BP53, 38041 Grenoble, Francee-mail: mickael.bourgoin@hmg.inpg.fr

The date of receipt and acceptance will be inserted by the editor

Abstract.Turbulent transport of material inclusions playsan important role in many natural and industrial situ-ations. Being able to accurately model and predict thedynamics of dispersed particles transported by a turbu-lent carrier flow, remains a challenge. One critical anddifficult point is to develop models which correctly de-scribe the dynamics of particles over a wide range of sizesand density. Our measurements show that accelerationstatistics of particles dispersed in a turbulent flow doexhibit specific, and so far unpredicted, size and densityeffects and that they preserve an extremely robust tur-bulent signature with lognormal fluctuations, regardlessof particles size and density. This has important conse-quences in terms of modeling for the turbulent transportof dispersed inclusions.

1 Introduction

Turbulent transport of material inclusions plays an im-portant role in many natural and industrial processes.In marine ecosystem for instance, interactions betweenthe turbulent sea and the gametes of marine animals isdeterminant for the efficiency of the reproduction andthe spreading of the species (1). Industrial stakes con-cern mixing and combustion, while environmental issuessuch as pollutants dispersion (2) and pollenization (3)are more than ever crucial. In spite of decades of activetheoretical, numerical and experimenal research, beingable to accurately model and predict the dynamics of dis-persed inclusions transported by a turbulent flow, stillremains a challenge with important scientific, environ-mental and economical issues.

One critical and difficult point is to find a modelwhich would correctly describe particles dynamics overa wide range of sizes and densities. The dynamics of par-ticles transported by a turbulent flow is indeed knownto be affected by size and density effects. If particles areneutrally buoyant and small (namely with their diameterD much smaller than the smallest turbulent eddies, at

the energy dissipation scale η of the flow), they behaveas fluid tracers and their dynamics reflects fluid parti-cles dynamics. This property is commonly used to char-acterize single phase flows from tracer particles imag-ing techniques (5) (such as Particle Image Velocimetry,Laser Doppler Velocimetry, Particle Tracking Velocime-try, etc.). However, when particles density is larger orsmaller than the surrounding fluid and/or when parti-cles size becomes comparable to turbulent eddies’, theirdynamics deviates from that of fluid particles and tendsto be affected by so called inertial effects. Among themsome can be qualitatively described in terms of particlesinteraction with turbulent eddies. For instance, centrifu-gal forces are generally expected to cluster heavier parti-cles in low vorticity regions (and lighter in high vorticityones), an effect known as preferential concentration (6).However, from a quantitative point of view no reliablemodel has emerged yet to describe and predict accu-rately the statistical properties of particles advected bya turbulent flow. For instance we are unable today to

40.429.221.015.210.97.95.7

4.1

3.0

2.1

1.5

1.1

0.8

10 12 14 16 18 20 22 24 26

100

101

102

φ = D / η

Γ =

ρp /

ρair

Fig. 1 Particle classes considered in the present study, de-scribed in the (φ, Γ ) phase space. φ = D/η is the ratio of par-ticles diameter to kolmogorov scale and Γ = ρp/ρair is theparticle to fluid density ratio. The countour lines indicate anestimation of the particles Stokes number St = τp/τD, wherethe respone time τp of the particle has been corrected foradded mass and for finite Reynolds number effects accordingto Schiller and Nauman (4) and τD is the flow eddy turnovertime at the scale of the particle diameter D.

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Author manuscript, published in "European Physical Journal B 66, 4 (2008) 531-536" DOI : 10.1140/epjb/e2008-00460-x

2 Qureshi et al.

xyz

U

receiver

measurement

volume

bubble

nozzle

mean flow

direction

2.75 m L = 75 cm

Data

Acq. ν0

ν0 + δν

Emitter

excitation

emitter

Fig. 2 Turbulence is generated downstream a grid in a wind tunnel. The measurement volume is located 2.75m downstreamthe grid, in a region where turbulence is fully developped, homogeneous and isotropic. Particles are injected individually atthe grid with a bubble nozzle and tracked individually by Doppler velocimetry. An ultrasonic plane wave of frequency ν0 isemitted by a transducer and a receiver records the wave scattered by a moving particle with velocity v in the measurementvolume (which is the intersection of the emmitting and receiving acoustical beams, represented by the light cloroed tubes onthe sketch). The component v‖ of the particle velocity along the scattering vector is directly given by the Doppler shift δν ofthe frequency of the recorded scattered wave. In the present experiment, transuducers are arranged in order to measure thesreamwise component vz of particles velocity.

predict correctly the dynamics of water droplets in acloud (7; 8), what considerably limits our understandingof rain mechanisms. Actually, even writing an appro-priate equation of motion for a particle transported bya turbulent flow remains a theoretical challenge whichhas only been approached in some limit cases, assuminggenerally point like particles (9; 10; 11; 12; 13; 14; 15).The range of validity of such models for real particles,with finite size and finite density, as well as the minimalforcing terms for a model to be pertinent remain un-clear. In this context, the study of particles Lagrangianacceleration, which directly reflects the turbulent forc-ing exerted on the particles, has been particularly ac-tive in the past decade (16; 17; 18; 19; 20; 21; 22). It isnow well known that fluid particles exhibit highly non-gaussian acceleration probability density function (PDF)with broad probability tails at high acceleration events(16). Direct numerical simulations in the limit of highlyinertial point particles suggest that acceleration PDFtends to narrow and to gaussianize as particles inertiaincreases. On the experimental side, studies in the limitof very small (D ≪ η) and highly inertial particles (103

denser than the carrier flow) seem to confirm this trend(18; 19). However, recent measurements with finite size(D & η) weakly inertial particles (up to 2.5 times denserthan the carrier flow) do not show any significant changeof acceleration PDF with inertia (21; 22).

In the present article, we report an exhaustive exper-imental investigation of material particles acceleration in

a turbulent air flow, over a wide range of sizes and den-sities. In particular we consider the so far unexploredregime of finite size (D > η) and highly inertial parti-cles (much denser than the carrier fluid). We consider aturbulent carrier flow generated downstream a grid in awind tunnel experiment.

The article is organized as follows : in section 2 wepresent the particles parameter space, which describesthe range of particles size and density we have effec-tively explored ; in section 3 we describe the wind tunnelsetup and the characteristic scales of the carrier turbu-lent field ; in section 4 we present the results obtainedconcerning particles acceleration statistics ; we finallypropose a discussion of these results in section 5 beforeconcluding in section 6.

2 Particles Parameter Space

The aim of the present work is to explore accelerationstatistics of inertial particles over a wide range of sizesand densities. In order to do so, we have developpeda versatile particle generator which produces in situ ad-justable spherical soap bubbles. Their density can be ad-justed from neutrally buoyant to about 70 times denserthan air. Neutrally buoyant particles are obtained by in-flating the bubbles with Helium in order to compensatethe weight of the soap film. Heavier particles are inflatedwith air, and their density is essentially controled by the

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Acceleration statistics of inertial particles in turbulent flow 3

-15 -10 -5 0 5 10 1510

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

az / < a

z

2 >

(1/2)

PD

F

12.5 114.6 116.7 118.8 1 20.8 1 22.3 1 25 1

13.5 6.34 15.3 4.67 16.7 4.58 18.5 5.9 20.8 4.23 22.8 4.0

16.7 2.7

16.8 12.6 16.8 30 12.4 47 16.4 50 19 52 21 41 16.8 65

0.05 1000

(from [18])

Lognormal

fit

φ Γ φ Γ

Fig. 3 Normalized component acceleration probability den-sity function (PDF) of material particles transported in a tur-bulent air flow. Dashed line corresponds to the measurementby Ayyalasomayajula et al. (18) for water droplets (with sizeφ ∼ 5 × 10−2 and Γ ∼ 1000). To help the comparison withprevious and future studies the dot-dashed line shows a fit by

the relation P(x) = e3s2/2

4√

3

»

1 − erf

„

ln(|x/√

3|)+2s2

√2s

«–

associ-

ated to a lognormal distribution of the acceleration amplitude(23; 20) ; best fit is found for s ∼ 0.62, corresponding to a

distribution flatness F = 9

5e4s2

∼ 8.3.

thickness of the soap film. In all cases the bubbles We-ber number remains extremely small, so that they do notdeform and behave as rigid spheres. The seeding densityis extremely low (particles are injected individually) sothat particles can be considered as isolated and do notbackreact on the carrier flow. Particles are therefore onlycharacterized by two parameters : the ratio φ of their di-ameter D to the dissipative scale of the flow (φ = D/η)and the ratio Γ of their density ρp to the carrier fluid (airin our case) density ρair (Γ = ρp/ρair). Figure 1 sum-marizes all the particle classes (in the (φ, Γ ) parameterspace) that we have investigated. As far as we know thisrepresents the most exhaustive exploration over such awide range of particles sizes and densities.

3 Experimental Setup

Our experiment runs in a large wind tunnel (figure 2)with a measurement section of 0.75 m × 0.75 m wherethe turbulence is generated downstream a grid with amesh size of 6 cm and reproduces almost ideal isotropicturbulence. The results reported here were obtained witha mean velocity of the fluid U = 15 m · s−1 and a turbu-lence level urms/U ≃ 3%. The corresponding Reynoldsnumber, based on Taylor microscale, is of the order ofRλ = 160. The dissipation scale η = (ν3/ǫ)1/4 is 240 µm(where ν is the kinematic viscosity of air and ǫ the tur-

bulent energy dissipation rate per unit mass) and the en-ergy injection scale L is 6 cm. The carrier turbulent fieldhas been characterized using classical hotwire anemom-etry. In particular, using a Taylor hypothesis, this givesaccess to the Eulerian correlation function and to theturbulent velocity gradients from which we estimate theintegral scale L and the dissipation rate ǫ respectively.

Particles are individually tracked using 1D Lagrangianacoustic Doppler velocimetry (see figure 2), where the in-stantaneous Doppler shift is obtained by a maximum oflikelyhood parametric algorithm (24; 25; 26). We mea-sure the streamwise velocity component vz of the par-ticles as they are tracked along their trajectories. Thestreamwiase acceleration component az is obtained bydifferentiating the velocity using a convolution with adifferentiated gaussian kernel (27). Each particle is trackedduring approximately 50 ms, which corresponds to sev-eral dissipation time scales τη = (ν/ǫ)1/2 ≃ 3.8 ms. Foreach point in the (φ, Γ ) parameter space we record about4000 tracks at a samplimg rate of 32768 Hz, giving morethan 106 data points per set.

4 Acceleration Statistics

Figure 3 represents acceleration PDFs normalized to vari-

ance one, (az/⟨

a2z

⟩1/2), for all studied particles. Remark-

ably we find that all the PDFs almost collapse onto asingle curve, indicating that, over the (φ, Γ ) range con-sidered, normalized acceleration statistics depends sig-nificantly neither on particle size, nor on particle density.In particular, contrary to predictions from point particlemodels, we do not observe any trend to gaussianization,even for particles with Stokes number St (see caption offigure 1 for the definition of St) as large as 40 ; withinstatistical errorbars, we do not observe any systematictrend of acceleration flatness, which keeps a constantvalue F ≃ 8.3 with 10% standard deviation among thedifferent particle classes. This result extends to the caseof heavy particles the robustness of acceleration PDFwhich we have previously reported for neutrally buoy-ant particles (20). It is also qualitatively consistent withrecent measurements, in water von Karman flows withweakly inertial particles (28; 21; 22), although an un-ambiguous quantitative comparison with those resultscannot be made due to experimental specificities (gridturbulence and von Karman flows have very differentlarge scale anisotropy and confinement characteristicswhich are known to affect particles Lagrangian dynamics(29; 30)). A quantitative comparison is however possi-ble with recent measurements obtained by Z. Warhaft’sgroup at Cornell University in a turbulent air flow withcomparable characteristics to ours (both are windtunnelexperiments with similar isotropy levels), although at aslightly higher Reynolds number (Rλ ∼ 250). Using highspeed optical tracking, they have measured accelerationstatistics of sub-kolmogorov water droplets (φ ∼ 5×10−2

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4 Qureshi et al.

-300 -200 -100 0 100 200 300

10-7

10-6

10-5

10-4

10-3

10-2

az (m/s

2)

PD

F

Γ ~ 1

Γ ~ 5

Fig. 4 Non normalized component acceleration PDF forΓ ∼ 1 (outer bundle, green tone) and Γ ∼ 5 (inner bun-dle, red tone) particles and various values of φ.

; Γ ∼ 1000 ; St ∼ 0.1), a regime of size which we can-not reach at present due to technical limitations of ourbubble generator. We have superimposed on figure 3 thenormalized acceleration PDF from their measurements(black dashed line). We find it to be almost undistin-guishable from our measurements with larger particles.This suggests that acceleration of inertial particles assmall as φ ∼ 5 × 10−2 already exhibit the same robuststatistical signature as finite size particles ; point par-ticle models may therefore only apply to even smallerparticles (19).

For a deeper insight into specific effects associatedto particles size and density, one has to investigate ac-celeration variance. Since the statistics of accelerationnormalized to variance one is found essentially indepen-dent of particles size and density, an effect of those pa-rameters can indeed only be expected to affect acceler-ation variance itself

⟨

a2z

⟩

. Figure 4 represents the trueacceleration PDF (not normalized to variance one) forneutrally buoyant (Γ ∼ 1) and heavy (Γ ∼ 5) particles,with varying sizes. It appears that heavy particles havea much narrower and peaked PDF, indicating a globaldecrease of acceleration variance with increasing density.The spreading of the curves for a given density indicatesthat size also influences particles acceleration variance.

A closer analysis of acceleration variance shows anon-trivial dependence with size and density. We reportin figure 5 the acceleration variance for all the mea-surements we have performed (in the following, we con-sider the dimensionless acceleration variance normalizedaccording to Heisenberg-Yaglom’s scaling : A0(φ, Γ ) =⟨

a2z

⟩

ǫ−3/2ν1/2). The mapping of the (φ, Γ ) parameterspace that we have been able to explore, allows us toinfer a rough interpolation of the evolution of A0(φ, Γ ),represented by the surface in figure 5. Though only the

Fig. 5 Acceleration variance for all particles studied in the(φ, Γ ) parameter space (dots). The surface represents a roughinterpolation based on the available measurements (a bi-harmonic spline interpolation (31) has been used). The dotdashed lines materialize the interpolation along the best re-solved constant φ-lines and Γ -lines. For the Γ = 1 set, thedot-dashed line coincide with a φ−2/3 decay for φ & 15(20; 16).

coarse tendency of this interpolation is relevant, severalimportant qualitative features can still be observed. If weconsider the effect of an increasing density at a fixed par-ticle size (along constant φ-lines), A0 is always found todecrease and to saturate to a finite limit (noted a∞

0(φ))

for the largest densities. This is better quantified in a 2Dprojection (figure 6). The φ ≃ 16.5 set of measurements(which is our most complete set of density effects at fixedsize) shows that the evolution of acceleration variancewith density exhibits two different regimes (see insetin figure 6): (i) for low densities A0(φ, Γ ) = a0(φ)Γ−α

(a0(φ) corresponds to size effects for the neutrally buoy-ant case which we have previously studied in (20)) withα ≃ 0.6 (ii) for large densities A0 saturates to a finitelimit, here a∞

0(φ ≃ 16.5) ≃ 0.7 ; the transition between

these two regimes occurs for a characteristic density ra-tio Γ ∗ ∼ 10. Although we don’t have as many system-atic measurements for varying densities for other valuesof φ, the few available points are compatible with theexistence of such two regimes (figure 6), with a size de-pendent saturation value a∞

0(φ) and very likely a size

dependent transition density ratio Γ ∗(φ).

If we focus now on size effects at fixed particle density(along constant Γ -lines), as seen on figure 5, the scenariois rather complex: depending on the density ratio Γ , A0

can either decrease or increase with particles size. Forneutrally buoyant particles (Γ = 1) larger than aboutφ ∼ 15, A0 decreases monotically as a0(φ) ∼ φ−2/3 (20;16). We have shown in a previous study (20) that thisis the expected scaling for inertial range sized parti-cles when the main forcing simply comes from the spa-tial pressure differences of the unperturbed flow around

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Acceleration statistics of inertial particles in turbulent flow 5

0 10 20 30 40 50 60 700

0.5

1

1.5

2

2.5

3

Γ

A0

φ ∼ 16.5

φ ∼ 13

φ ∼ 19

φ ∼ 21

100

101

100

Γ

A0

a∞(φ >19)0

a∞(φ < 17)0

Γ∗(φ < 17)

-0.6

~

~

~

Fig. 6 2D-Projection on the (Γ−A0) plane of measurementsdata points on figure 5, showing the evolution of accelera-tion variance with density ratio Γ for different particle sizes(φ ≃ 16.5 set (circles) is the best resolved). The inset showsthe same data in a log-log plot. Errorbars are mostly due toexperimental uncertainties in the determination of particlessize and density.

the particle. As we move to larger density ratios wethen observe a continuous transition toward a drasticallychanged size dependence where a∞

0(φ) exhibit a sharp

increase for sizes around 17 < φ < 19. Outside this tran-sition region (i.e. for φ . 17 and φ & 19), a∞

0doesn’t

exhibit significant dependence on φ (at least in the ac-cessible range of sizes) as also seen on figure 6, where wehave a∞

0(13) ≃ a∞

0(16.5) and a∞

0(19) ≃ a∞

0(21).

5 Discussion

Our measurements have important consequences in termsof the development of accurate models for the turbulenttransport of finite sized inertial particles. The robust-ness of normalized acceleration PDF, regardless of par-ticles size and density, and the non trivial size and den-sity dependence of acceleration variance impose strongconstraints on the statistical properties of the forcingterms to be included in the equation of motion of parti-cles. For instance models based on Maxey & Riley equa-tion (9; 10) for inertial point particles are contradictorywith our observations as they predict a continuous trendof acceleration PDF to gaussianity as well as a mono-tonic decrease of acceleration variance (13; 21; 32) bothmostly consequence of a filtering effect due to increasingparticles response time as inertia (only parametrized insuch models by a Stokes number) increases. Our findingstherefore show that turbulent transport of finite sizedinertial particles cannot be simply described in termsof Stokes number and filtering effects only. The con-crete statistical constraints that we have identified give afirst clear experimental diagnose of point-particle modelslimitations (already analyzed theoretically by Lumley in1978 (33)), and of the requirement for density and finite

size effects to be simultaneously and specifically mod-eled.

With this goal, finally, we briefly discuss a simplequalitative phenomenology for the observed trends ofacceleration variance based on the idea that particles

tend to reside longer in the quietest regions of the flow.This intuitive statement can be related to the tendencyof inertial particles to preferentially concentrate in lowvorticity and low acceleration regions of the carrier tur-bulent field along which they are then advected folowinga sweep-stick scenario (34; 35). In this context, the de-crease of particles acceleration variance naturally resultsfrom the under-sampling of intense regions of the carrierflow. Then, a possible interpretation for the increase ofacceleration variance with particle size observed for highdensity ratios can be that while small enough particlescan indeed be expected to efficiently hide in the quietestregions of the flow, when particles become larger thanthe typical size L∗ of these regions, the sampling effectis damped as particles start experiencing again the influ-ence of more active regions of the turbulent field. Thoughthe existence of such a typical scale is still controversial,recent measurements of preferential concentration sug-gest that the quiet sticking regions might have a char-acteristic size L∗ in the range 10 − 20η (36) consistentwith the 17 . φ . 19 range for which we observe thesudden increase of a∞

0(φ).

Our measurements of inertial effects for finite sizeparticles can therefore be consistently interpreted as adominant role of sampling rather than filtering, contraryto the point particle case (13; 32). The dominant role ofsampling is also consistent with the fact that velocitystatistics of our particles (not shown here) are foundidentical to that of the carrier flow (obtained from clas-sical hotwire anemometry), a feature also shown numer-ically in the context of sweep-stick mechanism (35) butwhich is at odds from usual predictions based for in-stance on the Tchen-Hinze theory or extensions of thistheory (37).

6 Conclusion

To summarize, using a versatile material particle gen-erator we have been able to explore the simultaneousinfluence of size and density on the turbulent transportof material particles. For fixed carrier flow conditions,acceleration statistics of finite size inertial particles arefound very robust to size and density variations, the in-fluence of which is mostly carried by acceleration vari-ance only. In particular the shape of acceleration proba-bility density function remains unchanged over the wholerange of sizes and densities explored. Concerning accel-eration variance, we find that, for heavy particles, finitesize effects can be trivially extrapolated neither fromthe heavy point particle limit nor from the finite sizeneutrally buoyant case as both of these limits predict

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a monotonic decrease of acceleration variance with in-creasing inertia, contradictory with the trends measuredfor finite size heavy particles. Observed finite size anddensity effects cannot be accounted by simple filteringarguments only based on Stokes number effects as in thepoint particle case. A sampling scenario, in the spiritof sweep-stick mechanisms seems more adequate to pos-sibly describe the simultaneous influence of density andsize. We believe this should stimulate further experimen-tal and numerical investigations of acceleration fields inturbulent flows as well as of inertial particles concen-tration fields. It is our hope that these measurementswill contribute bridging experimental observations anddevelopments of accurate models for the turbulent trans-port of particles.

Acknowledgements We deeply acknowledge E. Calzavarini,J.-F. Pinton, R. Volk, and Z. Warhaft for fruitful discussions.

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